Equational axioms for regular sets

1993 ◽  
Vol 3 (1) ◽  
pp. 1-24 ◽  
Author(s):  
S. L. Bloom ◽  
Z. Ésik
Keyword(s):  
Star Operation ◽  
Regular Sets

We show that, aside from the semiring equations, three equations and two equation schemes characterize the semiring of regular sets with the Kleene star operation.

scholarly journals Inductive * -Semirings

2000 ◽  
Vol 7 (27) ◽  
Author(s):  
Zoltán Ésik ◽  
Werner Kuich
Keyword(s):  
Fixed Point ◽  
Power Series ◽  
Computer Science ◽  
Equational Theory ◽  
Star Operation ◽  
Rational Power Series ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Regular Sets ◽  
Kleene Theorem

One of the most well-known induction principles in computer science<br />is the fixed point induction rule, or least pre-fixed point rule. Inductive <br />*-semirings are partially ordered semirings equipped with a star operation<br />satisfying the fixed point equation and the fixed point induction rule for<br />linear terms. Inductive *-semirings are extensions of continuous semirings<br />and the Kleene algebras of Conway and Kozen.<br />We develop, in a systematic way, the rudiments of the theory of inductive<br />*-semirings in relation to automata, languages and power series.<br />In particular, we prove that if S is an inductive *-semiring, then so is<br />the semiring of matrices Sn*n, for any integer n >= 0, and that if S is<br />an inductive *-semiring, then so is any semiring of power series S((A*)).<br />As shown by Kozen, the dual of an inductive *-semiring may not be inductive. <br />In contrast, we show that the dual of an iteration semiring is<br />an iteration semiring. Kuich proved a general Kleene theorem for continuous<br /> semirings, and Bloom and Esik proved a Kleene theorem for all Conway <br />semirings. Since any inductive *-semiring is a Conway semiring<br />and an iteration semiring, as we show, there results a Kleene theorem <br />applicable to all inductive *-semirings. We also describe the structure<br />of the initial inductive *-semiring and conjecture that any free inductive<br />*-semiring may be given as a semiring of rational power series with <br />coefficients in the initial inductive *-semiring. We relate this conjecture to<br />recent axiomatization results on the equational theory of the regular sets.

scholarly journals Prox-regular sets and Legendre-Fenchel transform related to separation properties

Optimization ◽  
2020 ◽  
pp. 1-33
Author(s):  
Samir Adly ◽  
Florent Nacry ◽  
Lionel Thibault
Keyword(s):  
Fenchel Transform ◽  
Regular Sets

scholarly journals Plenty of big projections imply big pieces of Lipschitz graphs

2021 ◽  
Author(s):  
Tuomas Orponen
Keyword(s):  
Lipschitz Graphs ◽  
Regular Sets ◽  
Big Pieces

AbstractI prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$ E ⊂ R d with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.

scholarly journals $$v_c$$-Noetherian domains and Krull domains

2021 ◽  
Author(s):  
Gyu Whan Chang
Keyword(s):  
Dedekind Domain ◽  
Closed Domain ◽  
One Dimensional ◽  
Star Operation ◽  
Krull Domain ◽  
Noetherian Domain ◽  
Integrally Closed ◽  
Krull Domains

AbstractLet D be an integrally closed domain, $$\{V_{\alpha }\}$$ { V α } be the set of t-linked valuation overrings of D, and $$v_c$$ v c be the star operation on D defined by $$I^{v_c} = \bigcap _{\alpha } IV_{\alpha }$$ I v c = ⋂ α I V α for all nonzero fractional ideals I of D. In this paper, among other things, we prove that D is a $$v_c$$ v c -Noetherian domain if and only if D is a Krull domain, if and only if $$v_c = v$$ v c = v and every prime t-ideal of D is a maximal t-ideal. As a corollary, we have that if D is one-dimensional, then $$v_c = v$$ v c = v if and only if D is a Dedekind domain.

Intersections of quotient rings and Prüfer v-multiplication domains

2016 ◽  
Vol 15 (08) ◽  
pp. 1650149 ◽  
Author(s):  
Said El Baghdadi ◽  
Marco Fontana ◽  
Muhammad Zafrullah
Keyword(s):  
Integral Domain ◽  
Algebraic Approach ◽  
Quotient Field ◽  
Topological Characterization ◽  
Locally Finite ◽  
Star Operation ◽  
Star Operations ◽  
Quotient Rings ◽  
Special Case

Let [Formula: see text] be an integral domain with quotient field [Formula: see text]. Call an overring [Formula: see text] of [Formula: see text] a subring of [Formula: see text] containing [Formula: see text] as a subring. A family [Formula: see text] of overrings of [Formula: see text] is called a defining family of [Formula: see text], if [Formula: see text]. Call an overring [Formula: see text] a sublocalization of [Formula: see text], if [Formula: see text] has a defining family consisting of rings of fractions of [Formula: see text]. Sublocalizations and their intersections exhibit interesting examples of semistar or star operations [D. D. Anderson, Star operations induced by overrings, Comm. Algebra 16 (1988) 2535–2553]. We show as a consequence of our work that domains that are locally finite intersections of Prüfer [Formula: see text]-multiplication (respectively, Mori) sublocalizations turn out to be Prüfer [Formula: see text]-multiplication domains (PvMDs) (respectively, Mori); in particular, for the Mori domain case, we reobtain a special case of Théorème 1 of [J. Querré, Intersections d’anneaux intègers, J. Algebra 43 (1976) 55–60] and Proposition 3.2 of [N. Dessagnes, Intersections d’anneaux de Mori — exemples, Port. Math. 44 (1987) 379–392]. We also show that, more than the finite character of the defining family, it is the finite character of the star operation induced by the defining family that causes the interesting results. As a particular case of this theory, we provide a purely algebraic approach for characterizing P vMDs as a subclass of the class of essential domains (see also Theorem 2.4 of [C. A. Finocchiaro and F. Tartarone, On a topological characterization of Prüfer [Formula: see text]-multiplication domains among essential domains, preprint (2014), arXiv:1410.4037]).

More on Regular Sets

1997 ◽  
pp. 19-24
Author(s):  
Dexter C. Kozen
Keyword(s):  
Regular Sets

More on Regular Sets

1977 ◽  
pp. 19-24
Author(s):  
Dexter C. Kozen
Keyword(s):  
Regular Sets
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